Purpose

To measure capacitance. To investigate the
exponential time dependence of charge accumulation and discharge on a
capacitor plate. To find the effective capacitance of the series and parallel
combinations of two different capacitors.

Background

The electric
force between two charges is very strong. The net charge of the universe is
zero and it is the natural state of a single charge to be as close as
possible to one of the opposite polarity. Of course, there can be instances
when the net charge in a particular area is nonzero.

What is a Capacitor?

A capacitor
presents a case where the charge is nonzero in a localized area. This
physical entity is an electrical energy storage device that is best
visualized by two parallel metallic plates separated by a given distance.
(Other capacitor configurations are used but the concepts laid out in the
following paragraphs for the two plates hold for different geometries.) A
battery or some other generator of electric potential difference is used to separate
tightly bound positive and negative charges, pushing charges of one sign to
one plate and charges of the other sign to the second plate. The amount of
charge that eventually accumulates on either of the plates is proportional to
the charging voltage V0. There is a nonzero charge on either plate but the net charge of the
capacitor (both plates) is still zero.

Capacitance

In general,
capacitors have different capacities to store charge, depending on the
geometry and the material between the plates. This charge storage capacity,
or capacitance, is
the constant of proportionality C between the magnitude Q of the charge on one of the plates and the
potential V across the plates. In equation form, this relationship is

(1)

The SI unit for
capacitance is thefarad, which is a coulomb per volt (1 F
= 1 C/1 V). The
unit we will use in this experiment is the microfarad (μF).

Charging and Discharging

Exponential Decay and the Time Constant

Simple enough measure the charge on a capacitor for
different charging voltages, plot the line and the slope is then the
capacitance, right? The experimental problem comes in measuring Q. Although charge-measuring instruments
exist, they are not useful for much else and are expensive and imprecise.

How then do we measure the capacitance? First look at how
the charge collects on a plate as a function of time.

Figure 1
RC
circuit for charging and discharging a capacitor.

We set up a circuit as in Fig. 1. Connect a resistor R
and a capacitor C in series to a voltage
source through a two-pole switch. This is called an RC series circuit.
Connecting the switch to terminal 1 charges the
capacitor, with the rate of the charging decreasing as time passes. At first,
the capacitor is easy to charge because there is very little charge on the
plates. But as charge accumulates on the plates, the voltage source must do
more work to move additional charges onto the plates because like charges
repel each other. Thus the capacitor charges in an exponential manner,
quickly at the beginning and more slowly as the capacitor gets closer to
being fully charged. If this connection is made at t = 0, the charge Q
on one of the plates after a time t is

(2)

Q0 is the magnitude of the maximum charge that accumulates on a plate.

Since the voltage
across the plates and the charge on one plate are proportional, we can write

(3)

Resistance
is an electrical property that impedes the flow of charges through a circuit.
It is measured in ohms
(signified by the Greek letter Ω). A series connection of two things means
that whatever goes through one has to go through the other.

When the
capacitor is fully charged and the switch is flipped to 2 at t = 0, there is an exponential decay of the
charge:

(4)

with the
time-dependent voltage being

(5)

The constant RC in the exponent of all these equations is
called the time constant. It is a measure of how fast the circuit reacts. It is also a number
that characterizes the exponential decay. Note that an ohm times a farad is a
second. (1
Ω × 1 F = 1 s)

We use either Eq.
3 or Eq. 5 to find the capacitance, by fitting voltage and time data to these
equations. Experimentally, we charge up a capacitor with a potential of V0while monitoring the voltage across the
capacitor as a function of time. An exponential fit of V0 − V(t) gives the time constant RC. If R is known, we can then calculate C. The discharge of a capacitor is easier
to analyze, since an exponential fit of V(t) should give the same time constant.

We use a signal
generator to electronically
flip the switch to 1 then 2 in Fig. 1 then back to 1 and again to
2. A signal
generator, clearly, is a voltage source that supplies a time-varying voltage.
(It's no longer DC, it's AC.) The symbol for this in a schematic is a
squiggle inside a circle. A voltmeter (something that measures voltage) is a
capital V inside a circle. The schematic of our test circuit for measuring
capacitance becomes Fig. 2.

Figure 2
Schematic of RC test circuit used in the lab.

Equivalent Capacitance of
Connected Capacitors

We can connect
two capacitors C1and C2in a simple circuit one of two ways:
either in series or in parallel. In either case, the two capacitors act in
the circuit like a single equivalent capacitance Ceq. What is the value of this capacitance?

Figure 3
Two capacitors
connected in parallel.

In the parallel
case (Fig. 3), there are two possible paths for charges to take coming from a
terminal of the power supply. The potential difference across each capacitor
is the same. Different amounts of charge collect on the two positive plates.
The net charge deposited on the combination is the sum of the two charges.
Thus the equivalent capacitance is the total charge divided by the potential
difference, or the sum of the individual capacitances.

(6)

Figure 4 Two capacitors in series.

In the series
case (Fig. 4), there is only one possible path for charges to follow and the
charge accumulation has to be the same on both capacitors. The potential
difference across both resistors is the sum of the voltages across each
separately. The reciprocal of the effective (equivalent) capacitance is the sum
of the reciprocals of the two separate values.

(7)

Pertinent Questions

There
are several objectives or questions to answer in this experiment. You must
address these points in the Experimental
Results and Conclusions section of your report. They are as follows:

1.Does the voltage across a discharging
capacitor (and hence the charge on a plate) decay exponentially?

2.If it is exponential, calculate the
capacitance and compare it to the value measured with a meter based on a
different technique.

3.Does the voltage decay in an exponential
manner according to Eq. 3 when the capacitor charges? Is the capacitance
calculated from this the same as when it discharges?

4.Measure a second capacitor of a different
nominal value, using the technique of analyzing the voltage decay on
discharge. Does it give the same result as the capacitance meter?

5.Connect these two capacitors in parallel and
measure their combined capacitance using the voltage decay technique. Is it
the same as the theoretical prediction of Eq. 6?

6.Connect the two capacitors in series and use
the decay technique to find the equivalent capacitance. Is it the same as
that predicted by Eq. 7?

Procedure

You need the following equipment:

wood-based circuit board with 20 junctions

Wavetek
27XT digital multimeter (DMM) for measuring capacitance (The displayed
value has an uncertainty of 5% of the reading +10 μF on the 2000 μF scale, as
specified by the manufacturer. An equivalent way of saying this is that the
accuracy is 5% +10.))

Capacitance from Meter

1.Turn
on the Wavetek DMM and turn the dial to the 2000 μF setting to read capacitance.

a.Use the banana plug leads to connect the
two sides of the 100 mF capacitor to the mA Cx Lx and COM inputs of the meter.

b.Give the meter 20 or 30 seconds to settle
down before recording the reading as C1.

c.Do the same thing for the 330 μF capacitor and record the value as C2.

d.Record the value of R ±
u{R} from the value on
the board.

Figure 5 Measuring the capacitance
with the DMM.

Computer Data Acquistion

2.You
need to construct the test circuit of Fig. 2. A picture of what it should
look like when you're done is shown in Fig. 6.

Figure 6
This is a photo of
the wiring and connections for the test circuit of Fig. 2. (The top capacitor
is not in the circuit.)

a.Connect the DIN plug of the Voltage Sensor
to Channel B of the ScienceWorkshop interface.

b. Connect a banana plug lead from the ~ jack
of the OUTPUT on the interface to the banana plug receptacle on the circuit
board shown in Fig. 6. (One end of the capacitor is connected to this.)

c.Connect a banana plug lead from the other
jack (the ground) to the bottom left plug on the board. (One side of the
resistor.)

d.Connect the red banana plug of the Voltage
Sensor to the ~ OUTPUT on the board, through the banana plug already there.

e.Connect the black lead of the Voltage
Sensor to the other side of the capacitor.

f.Connect a banana plug lead from the other
side of the capacitor to the other side of the resistor.

√Checkpoint! Have the TA check your circuit
before you proceed further.

3.Start
the computer program by clicking on Data Studio Experiments> Second
Quarter> Capacitors.
DataStudio starts up with a Workbook sheet that gives instructions and other
information on the experiment.

a.Double click on the Voltage Graph in the Displays menu of DataStudio to view the data as it
is acquired and press Start to begin data acquisition. The "switch" is turned on for
about a second then off for the same time and repeated. Voltage is measured
across the capacitor and plotted. Your screen should look like Fig. 7. If
nothing shows up or you have all negative voltages, you have a connection
problem. Consult the TA. Record the Run # and which capacitor you're using.

Figure 7 Voltage behavior across a capacitor on
discharging and charging.

b.Replace C1 with C2 and repeat the voltage measurement. Don't
forget to record the Run # and the capacitor. The plot will be superimposed
on the first graph.

c.Connect C1 and C2 in parallel as drawn in Fig. 3. (Your
assembled product should look something like Fig. 8.) Take voltage data for
this combination and record the Run # and what you're measuring.

Figure 8 Fig. 3 realized.

d.One more experimental run! Connect C1 and C2 in series as in Fig. 4. (Your assembled
product should look something like Fig. 9.) Take voltage data for this
combination. Record the Run # and the capacitor combination.

Figure 9 Fig. 4 realized.

√Checkpoint! Have the TA check the DataStudio graph
of all four data runs before you proceed further.

4.We
have to be able to get this data into Excel.

a.In DataStudio, click on File>
Export Data...> the
Run
# for C1 in the Voltage, ChB list> OK.

b.Save to the Desktop as a text file with a
suitably descriptive file name.

c.Go to the Desktop and right click on this
file.

d.Click on Open With Excel. An open file cannot be deleted by the
clearing routine run every half hour.

5.Repeat
this data transfer for the C2 data.

6.Transfer
data from DataStudio to the Desktop for the parallel combination data.

7.Repeat
for the series combination data. You should have four open Excel files.

Analysis

New units to remember:

1.capacitance: 1 F=1 C/V

2.capacitance: 1 μF=10-6 F

3.resistance: 1 Ω=1 s/F

Transferring Data to Excel

1.Open the Capacitors template,
click in the blue box, enter in your name(s), and save the file following the
standard naming conventions.

2.Go to the Excel file containing the C1 data.

3.This file contains 2000 points, most of which
you don't need. First you need to find V0, the
maximum voltage across the capacitor. (Yes, we are applying 4 volts across
it, but there's no uncertainty associated with that, so we'll check what we
measured.)

a.Go to the end of the data (hit Ctrl End).
This takes you to the last voltage value, for t=4 s.

b.Go one cell further down (B2004)
and click on the fx
by the formula bar to bring up the list of Excel functions. Find the MAX
function and choose it. A box will appear of Function Arguments,
with all the numbers in the column above already chosen.

c.Click OK. The maximum
value appears. Write this down on your data sheet.

4.The next task is to filter out the data for C1 discharging.

a.Scroll to where the time is 1.13 seconds. This
is about where the switch is first turned off and the voltage starts to decay.

b.Find the first point where the voltage is 40
or 50 mV less that the maximum.

c.Select all points (times and voltages),
including this one, down to where the voltage is just less than 0.2 V. (There
should be 70-80 points.)

d.Copy this selection.

e.Return to the ca file started from
the template.

f.Go to cell A22.

g.Paste the selected points.

5.Return to the C1 data file to retrieve the charging
data. The positive square wave put out by the Signal Generator has
a frequency of 0.4 Hz, meaning it has a period of 2.5 s; it is on for half
the period and off for the other half. So 1.25 s after the charging signal
goes to zero at t=1.13 s, it goes back to 4 V at t=1.13 s + 1.25 s = 2.38 s.

a.Scroll down until the time is 2.38. Voltages
just before this time should be zero, ± a few millivolts. Voltages just after
this should increase precipitously.

b.Find the first point after t=2.38 s where the
voltage is at least 0.020 V.

c.Select this point and all points after it where
the voltage is less than 3.8 V.

d.Copy this selection.

e.Return to the ca file started from
the template.

f.Go to cell F22.

g.Paste the selected points.

6.Go to the C2 Excel data file. We want the data
where it is discharging.

a.Scroll to where the time is 1.13 seconds.

b.Find the first point where the voltage is 40
or 50 mV less that the maximum.

c.Select all points down to where the voltage is
just less than 0.2 V.

d.Copy this selection.

e.Return to the ca file started from
the template.

f.Go to cell K22.

g.Paste the selected points.

7.Go to the parallel combination Excel file to
retrieve its discharging data. Repeat the procedure in Step 6, pasting the
data into P22.

8.From the series capacitor combination, we want
the discharging data pasted into U22.

9.In the ca file, enter in
the data for V0,
R,
u{R},
C1, and C2.

√Checkpoint! Have the TA check all the data you
have entered into Excel before you proceed further.

Calculations

10.Calculate the values of t − t0 for all
times for each data set. The first value should be zero for each.

11.Calculate the normalized voltages (V/V0) for each
of the four discharging data sets.

12.Calculate 1-V/V0 for the
charging data of C1.

13. For C1
Discharging, do an exponential least-squares fit using LOGEST
of the normalized voltage (as the y-value) vs. the time (as the x-value).
(Details on this function can be found in Appendix
3.)

a.Highlight the two cells just to the right of
the e-1/RC
and u{1/RC}
(1/s) labels.

b.Perform the LOGEST function
evaluation of V/V0vs.
t − t0. Set the two logical flags
to true, or one. Hit Shift/Control/Enter.
This gives you the two numbers indicated by the labels, for this data set.

c.Calculate RC = −1 / ln(e−1/RC)
.

d.Calculate the relative error in RC.
This is the second number from LOGEST times RC.

(8)

14.Calculate the capacitance in microfarads from
the values of R and RC.

15.Find the uncertainty (the absolute error) of
this calculation of C, given by

(9)

Note that you
have already calculated the value of u{RC}/RC and
that u{R} and R are given.

√Checkpoint! Have the TA check your calculations
before you proceed further.

16.Hold off on any comparisons for the moment.
Repeat the calculations in steps 13-15 for the other four data sets. (Use 1−V/V0 as the
y-value for C1
Charging.) To save keystrokes, use the power of Excel...

a.Highlight all the numbers from e−1/RC
through u{C}
(μF).

b.Copy this selection and paste into the
corresponding cells for the other four data sets. You are actually copying
the formulas, which use relative cell references, so they are good for each
data set, except for the LOGEST
ranges.

c.Correct the cell ranges for LOGEST
for the other four data sets.

Comparisons

17.Now for the theoretical values (with errors)
and the comparisons.

a.The theoretical value for the first capacitor, discharging, as well
as the second capacitor, discharging, is the capacitance meter value.Enter the meter
values into the appropriate cells, calculate the uncertainties, and compare
the experimental and meter values.

b.For the first capacitor, charging, use the value of the first
capacitor, discharging as the theoretical value.Remember that we
want to show that the exponential decay is characterized the same way,
whether the capacitor is charging or discharging.

c.For the two combinations of the two capacitors, the theoretical value
is calculated from Eq. 7 or 8. Use the experimental values of C1 and C2 in
these calculations, not the meter values.You are responsible for propagating
the uncertainties for both quantities.

Plots and Verifying Exponential
Dependence

18.Do a semilog plot of the five data sets (the
ones you did the least squares fits of) on the same graph.

a.Select the two columns for C1
Discharging.

b.Insert> Charts> Scatter> Scatter with
only Markers.

c.Chart Tools> Design> Move Chart> New
Sheet> OK.

d.Chart Layouts> Layout 9.

e.Change Chart and Axes Titles
appropriately.

f.Layout 9 automatically includes a Linear
Trendline. You want to make this Exponential. Right
click on the line and choose Format Trendline…> Exponential>
uncheck Display
equation on chart and Display R-squared value on chart>
Close.

i.The point of a graph is to tell at a glance if
the data fit the general theory. In this case we expect a straight line on
the semilog plot. If it is not a semilog plot, you will get a curve; it is
hard to tell if this curve is exponential or hyperbolic or some other strange
dependence.

j.Do the points for each plotted set fall pretty
much on a straight line when the y-axis is scaled logarithmically and the
x-axis scaled the good old-fashioned linear way?

k.Delete the references to the trendlines in the
legend.

Questions

1.Derive Eq. 9 (propagate the error for C).

2.Propagate the uncertainty of the equivalent
capacitance of two capacitors in series.